An important property of any vacuum character is that it must have unit leading coefficient, and that the remaining Fourier coefficients must be integral. Not every choice of MDE, i.e. of the real parameters which determine it, is compatible with such integrality. A first step in our classification program is to identify the MDEs which do give rise to a sensible vacuum character. In the context of 2d CFT, this idea goes back to the classic work of Mathur, Mukhi, and Sen [1]}, [2]}, [3]}, see e.g. [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}, [13]}, [14]}, [15]}, [16]}, [17]}, [18]}, [19]} for recent developments. As scanning over the set of real numbers parameterizing the MDE would clearly be impossible, our first step is to map the problem to a scan over integers. This can be done as follows:
